






"""simple docstring"""







from __future__ import annotations





def 		lowercase__(A   ,						A		)							->list[tuple[int, int]]:




						"""simple docstring"""



						lowercase__,      lowercase__							: Optional[int]=			position
						lowercase__							: Optional[Any]=			[
						    (y + 1, x + 2),
						    (y - 1, x + 2),
						    (y + 1, x - 2),
						    (y - 1, x - 2),
						    (y + 2, x + 1),
						    (y + 2, x - 1),
						    (y - 2, x + 1),
						    (y - 2, x - 1),
						]
						lowercase__							: List[str]=			[]

						for position in positions:
												lowercase__,      lowercase__							: Any=			position
												if 0 <= y_test < n and 0 <= x_test < n:
																		permissible_positions.append(UpperCamelCase__		)

						return permissible_positions





def 		lowercase__(A		)							->bool:




						"""simple docstring"""



						return not any(elem == 0 for row in board for elem in row		)





def 		lowercase__(A   ,						A   ,						A		)							->bool:




						"""simple docstring"""



						if is_complete(UpperCamelCase__		):
												return True

						for position in get_valid_pos(UpperCamelCase__   ,						len(UpperCamelCase__		)		):
												lowercase__,      lowercase__							: Optional[int]=			position

												if board[y][x] == 0:
																		lowercase__							: Dict=			curr + 1
																		if open_knight_tour_helper(UpperCamelCase__   ,						UpperCamelCase__   ,						curr + 1		):
																								return True
																		lowercase__							: Dict=			0

						return False





def 		lowercase__(A		)							->list[list[int]]:




						"""simple docstring"""



						lowercase__							: str=			[[0 for i in range(UpperCamelCase__		)] for j in range(UpperCamelCase__		)]

						for i in range(UpperCamelCase__		):
												for j in range(UpperCamelCase__		):
																		lowercase__							: Union[str, Any]=			1
																		if open_knight_tour_helper(UpperCamelCase__   ,						(i, j)   ,						1		):
																								return board
																		lowercase__							: int=			0

						lowercase__							: Union[str, Any]=			f'''Open Kight Tour cannot be performed on a board of size {n}'''
						raise ValueError(UpperCamelCase__		)


if __name__ == "__main__":
			import doctest

			doctest.testmod()







'''simple docstring'''




def __lowerCAmelCase					(				UpperCamelCase__							)      ->       bool:
    if not isinstance(UpperCamelCase__							,    UpperCamelCase__							):
        raise ValueError('''Input series is not valid, valid series - [2, 4, 6]'''							)
    if len(UpperCamelCase__							) == 0:
        raise ValueError('''Input list must be a non empty list'''							)
    if len(UpperCamelCase__							) == 1:
        return True
    __lowerCamelCase            =    series[1] - series[0]
    for index in range(len(UpperCamelCase__							) - 1							):
        if series[index + 1] - series[index] != common_diff:
            return False
    return True






def __lowerCAmelCase					(				UpperCamelCase__							)      ->       float:
    if not isinstance(UpperCamelCase__							,    UpperCamelCase__							):
        raise ValueError('''Input series is not valid, valid series - [2, 4, 6]'''							)
    if len(UpperCamelCase__							) == 0:
        raise ValueError('''Input list must be a non empty list'''							)
    __lowerCamelCase            =    0
    for val in series:
        answer += val
    return answer / len(UpperCamelCase__							)


if __name__ == "__main__":
  import doctest

  doctest.testmod()
